CPhill

For a statue of Miss Levans, the amount of gold required is proportional to the square of the height of the statue. A statue of Miss Levans that is 20 feet tall requires 80 pounds of gold. How many pounds of gold does a statue that is 25 feet tall require?

We need to find the proportinality constant, k

80  =  k * 20^2

80  = k * 400    divide  both sides by 400

1/5  = k

So

Lbs. required  =   k * 125^2  =  (1/5)(25)^2   =  1/5  * 625    =    125 lbs

Mmmm.....that's a lot  of stuff !!!....I'll do a few

Twelve hungry teenagers can devour a very large pizza in 20 minutes. How many teenagers would it take to finish a very large pizza in 15 minutes? (Assume that all teenagers eat pizza at the same rate.)

This is known as a "see-saw" problem

No. of teens  * 15  =   12 * 20

No. of teens * 15   =   240        divide both sides by 15

No. of teens   =   240 / 15    =    16

We can sole this with a proportion

Jim's Height                                                Flagpole Height

___________________               =          __________________

Shadow Length of Jim                             Flagpole Shadow Length   

  6                            Flagole Height

__             =            _____________ 

16                                72

[ 6 /16  =  3/8 ]...so....

3 / 8   =   Flagpole Height / 72             multiply both sides by 72

72 * 3  /  8   =  Flagpole Height

216 / 8  = Flagpole Height

27 ft  =  Flagpole Height

This is made more simple if we take numerator  and denominator separately

  4                 1

_____    +   ____

2x - 1            4

4 (4)            +      1(2x - 1)                  16 + 1 (2x - 1)

_______            ________     =      _____________    =

4(2x - 1)               4 (2x - 1)                 4 ( 2x - 1)

16 + 2x - 1                     15   + 2x

__________   =              _________     =    a / 1

  4 ( 2x - 1)                      4 ( 2x - 1)

Note that  we are now  dividing this result by  x + 1   which is the same as  (x + 1) / 1   =   b / 1

So.....we  have    [  a / 1  ]    ÷    [ b / 1]   =   [ a / 1 ]   x   [ 1 / b ]  =

15 + 2x                      1                      15  +  2x

_________   x      ______  =         _____________

  4 (2x - 1)              x + 1               4(2x - 1) ( x + 1)

x +  1/y  =  1

x  =  1 - 1/y  =      [ y - 1] / y

And

y + 1/z  =  1

1/z  =  1 - y    which implies that

z  =  1 / [ 1 - y]

So    x  y  z   =     

 ( [ y - 1] / y  )  * y *  [ 1 / ( 1 - y) ]    = 

[  y - 1 ]  *  [  1  / (1 - y) ]  =

-1 ( 1 - y)  *  [ 1 / (1 - y) ]  =

-1  

x      y

0     0

2     0

0     2

2     4

4     2

4     4

The numerator is a polynomial......so any real number  b will "work"

We only have to worry about the  denominator  being 0

To find which values make this so we have

10b^2 + 40b  =  0      factor

10b ( b + 4)  =  0

So  either   10b  = 0   and b  = 0   or  b + 4  = 0   and b  = -4

So...the excluded values  are   0  and - 4

The simplified rational function is  just as Rom found....

Heck.....tertre beat me to it  !!!!!

I think this might be correct.....

Call  the couples  AB   CD    EF

Each couple has 2 ways that they could sit  together  [ example, AB  or BA ]

And  we have 3!  ways to seat the couples themselves  

So....this gives  2^3 * 3!    =  48 ways that they could sit  together

And the total possible arrangements are 6!  = 720

So  the proability of a socially optimum configuration is just    48 / 720  =  1 / 15

-1  =  sqrt ( 3x - 2)  - sqrt (4x + 1)

sqrt (  4x+ 1) - 1  = sqrt (3x - 2)     square both sides

4x+ 1 -  2sqrt(4x + 1)  + 1  =  3x - 2   rearrange as

x + 4  =  2sqrt (4x + 1)    square both sides once more

x^2 + 8x + 16  = 4(4x + 1)

x^2 + 8x + 16 = 16x + 4

x^2 - 8x + 12  = 0    factor

(x - 6) ( x - 2)  = 0

Set each factor to 0 and solve for x and we have that  x = 6  or x = 2

Both solutions check out in the original equation !!!